Impatience is rising and the leadership is turning on itself. One camp wants Mrs May to sack Mr Johnson, who has been writing articles clearly dissenting from the government line; another camp wants to see the back of Philip Hammond, the chancellor, who is taking a cautious view of post-Brexit economic prospects.
"Turning on" is being used to mean "blame/betray" in this sense.
To turn on each other or to turn on oneself or itself=
The dog turned on his master. [bit him]
It means to fight with in your context, but can mean things like a dog biting a person too. They are fighting with each other.
